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%Investigating the Statistics of the Number of Dice-Rolls Until Reaching a Desired Goal
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{\bf 
Investigating the Statistics of the Number of Dice-Rolls Until Reaching a Desired Goal
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{\it Lucy MARTINEZ and Doron ZEILBERGER}
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\quad \quad\quad \quad\quad \quad{\it Everybody knows that the dice are loaded} - Leonard Cohen


{\bf Abstract}: We harness the great power of symbolic computation and experimental mathematics to investigate the
statistics of the number of (loaded) dice-rolls it takes until every face showed at least a certain desired number of times.
We also investigate much more general scenarios.



{\bf Preface}:  

In a recent fascinating article that appeared in the Mathematical Intelligencer, Simon Blatt, Uta Freiberg, and Vladimir Shikhman investigated
the mathematics of Talmudic Family Planning, where a couple has a predecided goal to ``Go forth and multiply'' until they have
begotten $n$ boys and $k$ girls, for some pre-decided $n$ and $k$. They stop as soon as they have reached {\it both} goals.
They  assumed that the probability of having a boy, $p$, is always the same, and the gender of each birth is independent of the others.
They were wondering about the expected family size, that they called $F(n,k,p)$ in terms of $n$, $k$, and $p$, and proved
the elegant {\it constant gender rule}, that asserts that if $B(n,k,p)$ and $G(n,k,p)$ are the expected number of boys
and expected number of girls, respectively (of course $F(n,k,p)=B(n,k,p)+G(n,k,p)$) then
$$
\frac{B(n,k,p)}{G(n,k,p)} \, = \ \frac{p}{1=p} \quad .
$$

Of course, mathematically this is equivalemt to tossing a coin whose probability of Heads is $p$ (and hence the probability of Tails is $(1-p)$) and stopping as soon
as you have encountered at least $n$ Heads and at least $k$ Tails. The minimum number of coin-tosses is $n+k$, but the total number is arbitrary.

In this paper we will generalize it to an {\bf arbitrary} die, with $k$ faces, let's call them $1, 2, \dots ,k$, where the probability
of landing on face $i$ is $p_i$, and of course
$$
\sum_{i=1}^{k} \, p_i \, = \, 1 \quad .
$$




{\bf References}


[AM] N

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Lucy Martinez and Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt  lm1154 at math dot rutgers dot edu} \quad, \quad {\tt DoronZeil at gmail dot com}   \quad .


Written: {\bf Oct. 15, 2025}.

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